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Passive Network Synthesis Revisited Malcolm C. Smith Department of Engineering University of Cambridge U.K. Mathematical Theory of Networks and Systems (MTNS) 17th International Symposium Kyoto International Conference Hall Kyoto, Japan, July 24-28, 2006 Semi-Plenary Lecture

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Passive Network Synthesis Revisited Malcolm C. Smith

Outline of Talk

• 1. Motivating example (vehicle suspension).
• 2. A new mechanical element.
• 3. Positive-real functions and Brune synthesis.
• 4. Bott-Duffin method.
• 5. Darlington synthesis.
• 6. Minimum reactance synthesis.
• 7. Synthesis of resistive n-ports.
• 8. Vehicle suspension.
• 9. Synthesis with restricted complexity.
• 10. Motorcycle steering instabilities.

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Passive Network Synthesis Revisited Malcolm C. Smith

Motivating Example – Vehicle Suspension Performance objectives

• 1. Control vehicle body in the face of variable loads.
• 2. Insulate effect of road undulations (ride).
• 3. Minimise roll, pitch under braking, acceleration and

cornering (handling).

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Passive Network Synthesis Revisited Malcolm C. Smith

Quarter-car Vehicle Model (conventional suspension) tyre load disturbances road disturbances mu ms damper spring kt

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Passive Network Synthesis Revisited Malcolm C. Smith

The Most General Passive Vehicle Suspension mu

Z(s)

ms kt Replace the spring and damper with a general positive-real impedance Z(s). But is Z(s) physically realisable?

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Passive Network Synthesis Revisited Malcolm C. Smith

Electrical-Mechanical Analogies

• 1. Force-Voltage Analogy.

voltage ↔ force current ↔ velocity Oldest analogy historically, cf. electromotive force.

• 2. Force-Current Analogy.

current ↔ force voltage ↔ velocity electrical ground ↔ mechanical ground Independently proposed by: Darrieus (1929), H¨ ahnle (1932), Firestone (1933). Respects circuit “topology”, e.g. terminals, through- and across-variables.

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Passive Network Synthesis Revisited Malcolm C. Smith

Standard Element Correspondences (Force-Current Analogy) v = Ri resistor ↔ damper cv = F v = L di

dt

inductor ↔ spring kv =

dF dt

C dv

dt

= i capacitor ↔ mass m dv

dt

= F v1 v1 v2 v2

i i F F

Electrical Mechanical

What are the terminals of the mass element?

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Passive Network Synthesis Revisited Malcolm C. Smith

The Exceptional Nature of the Mass Element Newton’s Second Law gives the following network interpretation of the mass element:

• One terminal is the centre of mass,
• Other terminal is a fixed point in the inertial frame.

Hence, the mass element is analogous to a grounded capacitor. Standard network symbol for the mass element: v1 = 0 v2 F

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Passive Network Synthesis Revisited Malcolm C. Smith

Table of usual correspondences

• Electrical

Mechanical

spring mass damper inductor capacitor resistor

i i i i i i v1 v1 v1 v1 v1

v1 = 0

v2 v2 v2 v2 v2 v2 F F F F F

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Passive Network Synthesis Revisited Malcolm C. Smith

Consequences for network synthesis Two major problems with the use of the mass element for synthesis of “black-box” mechanical impedances:

• An electrical circuit with ungrounded capacitors will not have a direct

mechanical analogue,

• Possibility of unreasonably large masses being required.

Question Is it possible to construct a physical device such that the relative acceleration between its endpoints is pro- portional to the applied force?

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Passive Network Synthesis Revisited Malcolm C. Smith

One method of realisation

terminal 2 terminal 1 gear rack pinions flywheel

Suppose the flywheel of mass m rotates by α radians per meter of relative displacement between the terminals. Then: F = (mα2) ( ˙ v2 − ˙ v1) (Assumes mass of gears, housing etc is negligible.)

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Passive Network Synthesis Revisited Malcolm C. Smith

The Ideal Inerter We define the Ideal Inerter to be a mechanical one-port device such that the equal and opposite force applied at the nodes is proportional to the relative acceleration between the nodes, i.e. F = b( ˙ v2 − ˙ v1). We call the constant b the inertance and its units are kilograms. The ideal inerter can be approximated in the same sense that real springs, dampers, inductors, etc approximate their mathematical ideals. We can assume its mass is small.

M.C. Smith, Synthesis of Mechanical Networks: The Inerter, IEEE Trans. on Automat. Contr., 47 (2002), pp. 1648–1662. MTNS 2006, Kyoto, 24 July, 2006 12

Passive Network Synthesis Revisited Malcolm C. Smith

A new correspondence for network synthesis

• Electrical

Mechanical

spring inerter damper inductor capacitor resistor

i i i i i i v1 v1 v1 v1 v1 v1 v2 v2 v2 v2 v2 v2

Y (s) = k

s dF dt = k(v2 − v1)

Y (s) = bs F = b d(v2−v1)

dt

Y (s) = c F = c(v2 − v1) Y (s) =

1 Ls di dt = 1 L(v2 − v1)

Y (s) = Cs i = C d(v2−v1)

dt

Y (s) = 1

R

i = 1

R(v2 − v1) F F F F F F

Y (s) = admittance = 1

impedance

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Passive Network Synthesis Revisited Malcolm C. Smith

Rack and pinion inerter made at Cambridge University Engineering Department mass ≈ 3.5 kg inertance ≈ 725 kg stroke ≈ 80 mm

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Passive Network Synthesis Revisited Malcolm C. Smith

Damper-inerter series arrangement with centring springs

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Passive Network Synthesis Revisited Malcolm C. Smith

Alternative Realisation of the Inerter

screw nut flywheel

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Passive Network Synthesis Revisited Malcolm C. Smith

Ballscrew inerter made at Cambridge University Engineering Department Mass ≈ 1 kg, Inertance (adjustable) = 60–180 kg

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Passive Network Synthesis Revisited Malcolm C. Smith

Electrical equivalent of quarter car model mu Fs

Y (s)

ms zr kt ˙ zr mu

Y (s)

ms Fs k−1

t

+ − Y (s) = Admittance =

Force Velocity = Current Voltage

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Passive Network Synthesis Revisited Malcolm C. Smith

Positive-real functions

• Definition. A function Z(s) is defined to be positive-real if one of the

following two equivalent conditions is satisfied:

• 1. Z(s) is analytic and Z(s) + Z(s)∗ ≥ 0 in Re(s) > 0.
• 2. Z(s) is analytic in Re(s) > 0, Z(jω) + Z(jω)∗ ≥ 0 for all ω at which

Z(jω) is finite, and any poles of Z(s) on the imaginary axis or at infinity are simple and have a positive residue.

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Passive Network Synthesis Revisited Malcolm C. Smith

Passivity Defined

• Definition. A network is passive if for all admissible v, i which are square

integrable on (−∞, T], T

−∞

v(t)i(t) dt ≥ 0.

• Proposition. Consider a one-port electrical network for which the impedance

Z(s) exists and is real-rational. The network is passive if and only if Z(s) is positive-real.

R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.

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Passive Network Synthesis Revisited Malcolm C. Smith

• O. Brune showed that any (ratio-

nal) positive-real function could be realised as the impedance

• r

admittance

• f

a network comprising resistors, capacitors, inductors and transformers. (1931)

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Minimum functions A minimum function Z(s) is a positive-real function with no poles or zeros

• n jR ∪ {∞} and with the real part of Z(jω) equal to 0 at one or more

frequencies. ReZ(jω) ω ω1 ω2

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Passive Network Synthesis Revisited Malcolm C. Smith

Foster preamble for a positive-real Z(s) Removal of poles/zeros on jR ∪ {∞}. e.g. s2 + s + 1 s + 1 = s + 1 s + 1 ↑ lossless ↓ s2 + 1 s2 + 2s + 1 =

• 2s

s2 + 1 + 1 −1 Can always reduce a positive-real Z(s) to a minimum function.

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle Let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0). Write L1 = X1/ω1 and Z1(s) = Z(s) − L1s. Case 1. (L1 < 0) Z(s) Z1(s) L1 < 0 (negative inductor!) Z1(s) is positive-real. Let Y1(s) = 1/Z1(s). Therefore, we can write Y2(s) = Y1(s) − 2K1s s2 + ω2

1

for some K1 > 0.

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) Then: Z(s) Z2(s) L1 < 0 L2 > 0 C2 > 0 where L2 = 1/2K1, C2 = 2K1/ω2

1 and Z2 = 1/Y2.

Straightforward calculation shows that Z2(s) = sL3+ Z3(s) ↑ proper where L3 = −L1/(1 + 2K1L1). Since Z2(s) is positive-real, L3 > 0 and Z3(s) is positive-real.

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) Then: Z(s) Z3(s) L1 < 0 L2 > 0 L3 > 0 C2 > 0 To remove negative inductor: L1 L3 M Lp L2 Ls Lp = L1 + L2 Ls = L2 + L3 M = L2 Some algebra shows that: Lp, Ls > 0 and

M2 LpLs = 1 (unity coupling

coefficient).

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) Realisation for completed cycle: M Lp Ls Z(s) Z3(s) C2

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) Case 2. (L1 > 0). As before Z1(s) = Z(s) − L1s Z(s) Z1(s) L1 > 0 (no need for negative inductor!) Problem: Z1(s) is not positive-real! Let’s press on and hope for the best!! As before let Y1 = 1/Z1 and write Y2(s) = Y1(s) − 2K1s s2 + ω2

1

. Despite the fact that Y1 is not positive-real we can show that K1 > 0.

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) Hence: Z(s) Z2(s) L1 > 0 L2 > 0 C2 > 0 But still Z2(s) is not positive-real. Again we can check that Z2(s) = sL3+ Z3(s) ↑ proper where L3 = −L1/(1 + 2K1L1). This time L3 < 0 and Z3(s) is positive-real.

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Passive Network Synthesis Revisited Malcolm C. Smith

The Brune cycle (cont.) So: Z(s) Z3(s) L1 > 0 L2 > 0 L3 < 0 C2 > 0 As before we can transform to: M Lp Ls Z(s) Z3(s) C2 where Lp, Ls > 0 and

M2 LpLs = 1 (unity coupling coefficient).

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Passive Network Synthesis Revisited Malcolm C. Smith

• R. Bott and R.J. Duffin showed

that transformers were un- necessary in the synthesis of positive-real functions. (1949)

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Passive Network Synthesis Revisited Malcolm C. Smith

Richards’s transformation

• Theorem. If Z(s) is positive-real then

R(s) = kZ(s) − sZ(k) kZ(k) − sZ(s) is positive-real for any k > 0. Proof. Z(s) is p.r. ⇒ Y (s) = Z(s) − Z(k) Z(s) + Z(k) is b.r. and Y (k) = 0 ⇒ Y ′(s) = k + s k − sY (s) is b.r. ⇒ Z′(s) = 1 + Y ′(s) 1 − Y ′(s) is p.r. R(s) = Z′(s) after simplification.

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Passive Network Synthesis Revisited Malcolm C. Smith

Bott-Duffin construction (cont.) Idea: use Richards’s transformation to eliminate transformers from Brune cycle. As before, let Z(s) be a minimum function with Z(jω1) = jX1 (ω1 > 0). Write L1 = X1/ω1. Case 1. (L1 > 0) Since Z(s) is a minimum function we can always find a k s.t. L1 = Z(k)/k. Therefore: R(s) = kZ(s) − sZ(k) kZ(k) − sZ(s) has a zero at s = jω1.

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Passive Network Synthesis Revisited Malcolm C. Smith

Bott-Duffin construction (cont.) We now write: Z(s) = kZ(k)R(s) + Z(k)s k + sR(s) = kZ(k)R(s) k + sR(s) + Z(k)s k + sR(s) = 1

1 Z(k)R(s) + s kZ(k)

+ 1

k Z(k)s + R(s) Z(k)

.

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Passive Network Synthesis Revisited Malcolm C. Smith

Bott-Duffin construction (cont.) Z(s) = 1

1 Z(k)R(s) + s kZ(k)

+ 1

k Z(k)s + R(s) Z(k) Z(k) k

Z(s)

kZ(k)

Z(k)R(s)

Z(k) R(s)

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Passive Network Synthesis Revisited Malcolm C. Smith

Bott-Duffin construction (cont.) We can write:

1 Z(k)R(s) = const × s s2+ω2

1 +

1 R1(s) etc.

R1 R2 Z(s)

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Passive Network Synthesis Revisited Malcolm C. Smith

Example — restricted degree

• Proposition. Consider the real-rational function

Yb(s) = k a0s2 + a1s + 1 s(d0s2 + d1s + 1) where d0, d1 ≥ 0 and k > 0. Then Yb(s) is positive real if only if the following three inequalities hold: β1 := a0d1 − a1d0 ≥ 0, β2 := a0 − d0 ≥ 0, β3 := a1 − d1 ≥ 0.

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Passive Network Synthesis Revisited Malcolm C. Smith

Brune Realisation Procedure for Yb(s) Foster preamble always sufficient to complete the realisation if β1, β2 > 0. (No Brune or Bott-Duffin cycle is required). A continued fraction expansion is obtained: Yb(s) = k a0s2 + a1s + 1 s(d0s2 + d1s + 1) = k s + 1 s kb + 1 c3 + 1 1 c4 + 1 b2s

where kb = kβ2 d0 , c3 = kβ3, c4 = kβ4 β1 , b2 = kβ4 β2 and β4 := β2

2 − β1β3.

k c3 b2 kb c4

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Passive Network Synthesis Revisited Malcolm C. Smith

Darlington Synthesis Realisation in Darlington form: a lossless two-port terminated in a single resistor. I1 I2 V1 V2 R Ω Y (s)

Lossless network

For a lossless two-port with impedance: Z =   Z11 Z12 Z12 Z22   we find Z1(s) = Z11 R−1(Z11Z22 − Z2

12)/Z11 + 1

R−1Z22 + 1 .

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Passive Network Synthesis Revisited Malcolm C. Smith

Writing Z1 = m1 + n1 m2 + n2 = n1 m2 m1/n1 + 1 n2/m2 + 1, where m1, m2 are polynomials of even powers of s and n1, n2 are polynomials

• f odd powers of s, suggests the identification:

Z11 = n1 m2 , Z22 = R n2 m2 , Z12 = √ R √n1n2 − m1m2 m2 . Augmentation factors are necessary to ensure a rational square root. Once Z(s) has been found, we then write: Z(s) = sC1 + s s2 + α2 C2 + · · · where C1 and C2 are non-negative definite constant matrices.

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Passive Network Synthesis Revisited Malcolm C. Smith

Darlington Synthesis (cont.) Each term in the sum is realised in the form of a T-circuit and a series connection of all the elementary two-ports is then made: X1(s) X2(s) X3(s) 1: n

Ideal Network 1 Network 2

I = 0

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Passive Network Synthesis Revisited Malcolm C. Smith

Electrical and mechanical realisations of the admittance Yb(s) k−1

2

H k−1

3

H 1: − ρ k−1

1

H b F R1 Ω k2 k3 λ1 λ2 k1 b c

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Passive Network Synthesis Revisited Malcolm C. Smith

Minimum reactance synthesis Z(s) Nondynamic Network X sL1

1 sC1

Let L1 = · · · = C1 = · · · = 1. If M =   M11 M12 M21 M22   is the hybrid matrix of X, i.e.   v1 i2   = M   i1 v2   , then Z(s) = M11−M12(sI+M22)−1M21.

D.C. Youla and P. Tissi, “N-Port Synthesis via Reactance Extraction, Part I”, IEEE International Convention Record, 183–205, 1966.

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Passive Network Synthesis Revisited Malcolm C. Smith

Minimum reactance synthesis Conversely, if we can find a state-space realisation Z(s) = C(sI − A)−1B + D such that the constant matrix M =   D −C B −A   has the properties M + M ′ ≥ 0, diag{I, Σ}M = Mdiag{I, Σ} where Σ is a diagonal matrix with diagonal entries +1 or −1. Then M is the hybrid matrix of the nondynamic network terminated with inductors or capacitors, which realises Z(s). A construction is possible using the positive-real lemma and matrix factorisations.

B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, 1973.

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Passive Network Synthesis Revisited Malcolm C. Smith

Synthesis of resistive n-ports Let R be a symmetric n × n matrix. A necessary and sufficient condition for R to be realisable as the driving-point impedance of a network comprising resistors and transformers only is that it is non-negative definite. No necessary and sufficient condition is known in the case that transformers are not available. A general necessary condition is known: that the matrix is paramount.1 A matrix is defined to be the paramount if each principal minor of the matrix is not less than the absolute value of any minor built from the same rows. It is also known that paramountcy is sufficient for the case of n ≤ 3.2

• 1I. Cederbaum, “Conditions for the Impedance and Admittance Matrices of n-ports without

Ideal Transformers”, IEE Monograph No. 276R, 245–251, 1958.

• 2P. Slepian and L. Weinberg, ”Synthesis applications of paramount and dominant matrices”,
• Proc. National Electron. Conf., vol. 14, Chicago, Illinois, Oct. 611-630, 1958.

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Passive Network Synthesis Revisited Malcolm C. Smith

Simple Suspension Struts k c b k c b k c b k c k1 k2 Layout S1 Layout S2 Layout S3 Layout S4 parallel series

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Passive Network Synthesis Revisited Malcolm C. Smith

Performance Measures Assume: Road Profile Spectrum = κ|n|−2 (m3/cycle) where κ = 5 × 10−7 m3cycle−1 = road roughness parameter. Define: J1 = E

• ¨

z2

s(t)

• ride comfort

= r.m.s. body vertical acceleration

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Passive Network Synthesis Revisited Malcolm C. Smith

Optimisation of J1 (ride comfort)

1 2 3 4 5 6 7 8 9 10 11 12 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

static stiffness J1

1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18

static stiffness % J1 (a) Optimal J1 (b) Percentage improvement in J1 Key: layout S1 (bold), layout S2 (dashed), layout S3 (dot-dashed), and layout S4 (solid).

M.C. Smith and F-C. Wang, 2004, Performance Benefits in Passive Vehicle Sus- pensions Employing Inerters, Vehicle System Dynamics, 42, 235–257. MTNS 2006, Kyoto, 24 July, 2006 48

Passive Network Synthesis Revisited Malcolm C. Smith

Control synthesis formulation

F F

Fs

K(s)

zr zs zu mu kt ms ks z w G(s) K(s) F ˙ zs − ˙ zu Ride comfort: w = zr, z = ˙ zs Performance measure: J1 = const. × Tˆ

zr→sˆ zs2

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Passive Network Synthesis Revisited Malcolm C. Smith

Bilinear Matrix Inequality (BMI) formulation Let K(s) = Ck(sI − Ak)−1Bk + Dk and Tˆ

zr→sˆ zs = Ccl(sI − Acl)−1Bcl.

• Theorem. There exists a positive real controller K(s) such that

Tˆ

zr→sˆ zs2 < ν and Acl is stable, if and only if the following problem is

feasible for some Xcl > 0, Xk > 0, Q, ν2 and Ak, Bk, Ck, Dk of compatible dimensions:  AT

clXcl + XclAcl

XclBcl BT

clXcl

−I  < 0,  Xcl CT

cl

Ccl Q  > 0, tr(Q) < ν2,  AT

k Xk + XkAk

XkBk − CT

k

BT

k Xk − Ck

−DT

k − Dk

  < 0.

• C. Papageorgiou and M.C. Smith, 2006, Positive real synthesis using matrix inequalities for mechanical

networks: application to vehicle suspension, IEEE Trans. on Contr. Syst. Tech., 14, 423–435. MTNS 2006, Kyoto, 24 July, 2006 50

Passive Network Synthesis Revisited Malcolm C. Smith

A special problem What class of positive-real functions Z(s) can be realised using one damper,

• ne inerter, any number of springs and no transformers?

Z(s) X c b Leads to the question: when can X be realised as a network of springs?

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Passive Network Synthesis Revisited Malcolm C. Smith

• Theorem. Let

Y (s) = (R2R3 − R2

6) s3 + R3 s2 + R2 s + 1

s(det R s3 + (R1R3 − R2

5) s2 + (R1R2 − R2 4) s + R1),

(1) where R :=     R1 R4 R5 R4 R2 R6 R5 R6 R3     is non-negative definite. A positive-real function Y (s) can be realised as the driving-point admittance

• f a network comprising one damper, one inerter, any number of springs and

no transformers if and only if Y (s) can be written in the form of (1) and there exists an invertible diagonal matrix D = diag{1, x, y} such that DRD is paramount. An explicit set of inequalities can be found which are necessary and sufficient for the existence of x and y.

M.Z.Q. Chen and M.C. Smith, Mechanical networks comprising one damper and one inerter, in preparation. MTNS 2006, Kyoto, 24 July, 2006 52

Passive Network Synthesis Revisited Malcolm C. Smith

Collaboration with Imperial College Application to motorcycle stability. At high speed motorcycles can experience significant steering instabilities. Observe: Paul Orritt at the 1999 Manx Grand Prix

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Passive Network Synthesis Revisited Malcolm C. Smith

Weave and Wobble oscillations Steering dampers improve wobble (2–4 Hz) and worsen weave (6–9 Hz). Simulations show that steering inerters have, roughly, the opposite effect to the damper. Root-loci (with speed the varied parameter):

Steering damper root-locus Steering inerter root-locus

−18 −16 −14 −12 −10 −8 −6 −4 −2 2 4 10 20 30 40 50 60

Real Axis Imaginary Axis wobble weave ✻

−15 −10 −5 5 10 20 30 40 50 60

Real Axis Imaginary Axis wobble weave ✻

Can the advantages be combined?

• S. Evangelou, D.J.N. Limebeer, R.S. Sharp and M.C. Smith, 2006, Steering compensation for high-

performance motorcycles, Transactions of ASME, J. of Applied Mechanics, 73, to appear. MTNS 2006, Kyoto, 24 July, 2006 54

Passive Network Synthesis Revisited Malcolm C. Smith

Solution — a steering compensator ... consisting of a network of dampers, inerters and springs. Needs to behave like an inerter at weave frequencies and like a damper at wobble frequencies. damper inerter

Prototype designed by N.E. Houghton and manufactured in the Cambridge University Engineering Department.

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Passive Network Synthesis Revisited Malcolm C. Smith

Conclusion

• A new mechanical element called the “inerter” was introduced which is

the true network dual of the spring.

• The inerter allows classical electrical network synthesis to be mapped

exactly onto mechanical networks.

• Applications of the inerter: vehicle suspension, motorcycle steering and

vibration absorption.

• Economy of realisation is an important problem for mechanical network

synthesis.

• The problem of minimal realisation of positive-real functions remains

unsolved.

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Acknowledgements Research students Michael Chen, Frank Scheibe Design Engineer Neil Houghton Technicians Alistair Ross, Barry Puddifoot, John Beavis Collaborations Fu-Cheng Wang (National Taiwan University) Christos Papageorgiou (University of Cyprus) David Limebeer, Simos Evangelou, Robin Sharp (Imperial College)

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